Cluster validity measures for data with tolerance

Cluster validity measures are used in order to determine an appropriate number of clusters and evaluate cluster partitions obtained by clustering algorithms. When we handle a set of data, data contains inherent uncertainty e.g., errors, ranges or some missing value of attributes. The concept of tolerance has been proposed from the viewpoint of handling such uncertain data. In this paper, we introduce clustering algorithms for data with tolerance. Moreover, we propose new five measures for data with tolerance, that is, the determinants and the traces of fuzzy covariance matrices, the Xie-Beni's index, the Fukuyama-Sugeno's index, and the Davies-Bouldin's index. We compare the performance of conventional ones with their tolerance versions. We found that our proposed measures takes smaller value than conventional ones. These results indicate tolerance based clustering method is suitable for handling uncertain data.